Object: The angles of arterial bifurcations are governed by principles of work minimization (optimality principle). This determines the relationship between the angle of a bifurcation and the radii of the vessels. Nevertheless, the model is predicated on an absence of significant communication between these branches. The circle of Willis changes this relationship because the vessels proximal to the ring of vessels have additional factors that determine work minimization compared with more distal branches. This must have an impact on understanding of the relationship between shear stress and aneurysm formation. The authors hypothesized that normal bifurcations of cerebral arteries beyond the circle of Willis would follow optimality principles of minimum work and that the presence of aneurysms would be associated with deviations from optimum bifurcation geometry. Nevertheless, the vessels participating in (or immediately proximal to) the circle of Willis may not follow the geometric model as it is generally applied and this must also be investigated.
Methods: One hundred seven bifurcations of the middle cerebral artery (MCA), distal internal carotid artery (ICA), and basilar artery (BA) were studied in 55 patients. The authors analyzed three-dimensional reconstructions of digital subtraction angiography images with respect to vessel radii and bifurcation angles. The junction exponent (that is, a calculated measure of the division of flow at the bifurcation) and the difference between the predicted optimal and observed branch angles were used as measures of deviation from the geometry thought best to minimize work. The mean junction exponent for MCA bifurcations was 2.9 +/- 1.2 (mean +/- standard deviation [SD]), which is close to the theoretical optimum of 3, but it was significantly smaller (p < 0.001; 1.7 +/- 0.8, mean +/- SD) for distal ICA bifurcations. In a multilevel multivariate logistic regression analysis, only the observed branch angles were significant independent predictors for the presence of an aneurysm. The odds ratio (OR) (95% confidence interval) for the presence of an aneurysm was 3.46 (1.02-11.74) between the lowest and highest tertile of the observed angle between the parent vessel and the largest branch. The corresponding OR for the smallest branch was 48.06 (9.7-238.2).
Conclusions: The bifurcation beyond the circle of Willis (that is, the MCA) closely approximated optimality principles, whereas the bifurcations within the circle of Willis (that is, the distal ICA and BA) did not. This indicates that the confluence of hemodynamic forces plays an important role in the distribution of work at bifurcations within the circle of Willis. In addition, the observed branch angles were predictors for the presence of aneurysms.